4x2+14x+15=0

To solve the quadratic equation \(4x^2 + 14x + 15 = 0\), we can use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In this equation, the coefficients are:

• \(a = 4\)
• \(b = 14\)
• \(c = 15\)

Now, we can calculate the discriminant \(b^2 - 4ac\):

\[ b^2 = 14^2 = 196 \] \[ 4ac = 4 \cdot 4 \cdot 15 = 240 \] \[ b^2 - 4ac = 196 - 240 = -44 \]

Since the discriminant is negative (\(-44\)), the equation has no real solutions, only complex solutions.

Now we can calculate the complex solutions:

\[ x = \frac{-14 \pm \sqrt{-44}}{2 \cdot 4} \]

\[ \sqrt{-44} = \sqrt{44}i = \sqrt{4 \cdot 11}i = 2\sqrt{11}i \]

Putting this into the quadratic formula gives:

\[ x = \frac{-14 \pm 2\sqrt{11}i}{8} \]

We can simplify this:

\[ x = \frac{-14}{8} \pm \frac{2\sqrt{11}i}{8} \] \[ x = -\frac{7}{4} \pm \frac{\sqrt{11}}{4}i \]

So the solutions to the equation \(4x^2 + 14x + 15 = 0\) are:

\[ x = -\frac{7}{4} + \frac{\sqrt{11}}{4}i \quad \text{and} \quad x = -\frac{7}{4} - \frac{\sqrt{11}}{4}i \]